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Unformatted text preview: Chapter 1 Geometric Transformations Topics : 1. The Euclidean Plane E 2 2. Transformations 3. Properties of Transformations a114 Copyright c circlecopyrt Claudiu C. Remsing, 2006. All rights reserved. 1 2 M2.1  Transformation Geometry 1.1 The Euclidean Plane E 2 Consider the Euclidean plane (or twodimensional space ) E 2 as studied in high school geometry. Note : It is customary to assign different meanings to the terms set and space . Intuitively, a space is expected to possess a kind of arrangement or order that is not required of a set. The necessity of a structure in order for a set to qualify as a space may be rooted in the feeling that a notion of “proximity” (in some sense not necessarily quantitative) is inherent in our concept of a space. Thus a space differs from the mere set of its elements by possessing a structure which in some way (however vague) gives expression to that notion. A direct quantitative measure of proximity is introduced on an abstract set S by associating with each ordered pair ( x, y ) of its elements (called “points”) a nonnegative real number, denoted by d ( x, y ), and called the “distance” from x to y . On this “geometric space” one introduces Cartesian coordinates which are used to define a onetoone correspondence P mapsto→ ( x P , y P ) between E 2 and the set R 2 of all ordered pairs of real numbers. This mapping preserves distances between points of E 2 and their images in R 2 . It is the existence of such a coordinate system which makes the identification of E 2 and R 2 possible. Thus we can say that R 2 may be identified with E 2 plus a coordinate system . Note : The geometers before the 17th century did not think of the Euclidean plane E 2 as a “space” of ordered pairs of real numbers. In fact it was defined axiomatically beginning with undefined objects such as points and lines together with a list of their properties – the axioms – from which the theorems of geometry where then deduced. The identification of E 2 and R 2 (or, more generally, of E n and R n ) came about after the invention of analytic geometry by P. Fermat (16011665) and R. C.C. Remsing 3 Descartes (15961650) and was eagerly seized upon since it is very tricky and dif ficult to give a suitable definition of Euclidean space (of any dimension) in the spirit of Euclid. This difficulty was certainly recognized for a long time, and has inter ested many great mathematicians. It lead in part to the discovery of nonEuclidean geometries (like spherical and hyperbolic geometries) and thus to manifolds . We make the following definition. 1.1.1 Definition. The Euclidean plane E 2 is the set R 2 together with the Euclidean distance between points P = ( x P , y P ) and Q = ( x Q , y Q ) given by d ( P, Q ) = PQ : = radicalBig ( x Q x P ) 2 + ( y Q y P ) 2 ....
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 Fall '11
 y.talu
 Geometry, Transformations

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